Direct Sum Question

• October 20th 2011, 05:33 AM
KelvinScale
Direct Sum Question
Hi everyone,

I am having a bit of difficulty finding an example that works for this question. It asks to find subspaces W1, W2, and K, in vector space V such that

V = K *direct sum* W1
V = K *direct sum* W2

but W1 does not equal W2.

I have considered using K as R^n, and W1 as 0, and W2, as one vector in R^n. however, K and W2 have a non-zero intersection. Am I thinking in the wrong vector space?
• October 20th 2011, 07:41 AM
FernandoRevilla
Re: Direct Sum Question
Choose for example the subspaces of $V=\mathbb{R}^2$:

$W_1=\{(\lambda,0):\lambda\in \mathbb{R}\},\;W_2=\{(0,\mu):\mu\in \mathbb{R}\},\;K=\{(\gamma,\gamma):\gamma\in \mathbb{R}\}$